The two words have a similar meaning and I'm having difficulty adequately explaining the difference to someone. You verify a person's identity by having them show a form of identification, such as a passport of driver's license. You also check for signs that the document is not a forgery. If it all checks out, you'd let them into a bar, let them open a bank account or something like that.
Authentication is very similar. You ask for knowledge, possession and/or inherence factors to check if someone for example is allowed to log in to his e-mail or internet banking. Is there a difference between the two, and if so, what's an easy to understand explanation for it?
Edit to clarify what's confusing for me: isn't the verification of a person's identity in my example a form of authentication with a possession factor?